Chapter 3: Teleportation

Here is the complete question along with its detailed solution:

Problem Description

The goal of this problem is to analyze and demonstrate a quantum teleportation protocol involving three parties: Alice, Bob, and Charlie. The task is to verify whether a shared entangled state can be successfully transferred from Alice to Bob and Charlie, ensuring that the final state is preserved.

Initial Quantum State of Alice, Bob, and Charlie

We begin with a three-qubit system, where Alice initially possesses two qubits (denoted as \( A1 \) and \( A2 \)), while Charlie and Bob each have one qubit (denoted as \( C \) and \( B \), respectively).

• Alice has a two-qubit entangled state \( |\Psi\rangle_{A1C} \) with Charlie.
• Bob and Alice also share a maximally entangled Bell state \( |\phi^+\rangle_{A2B} \).

Thus, the initial state before the protocol starts is given by:

\[ |\Psi\rangle_{A1C} \otimes |\phi^+\rangle_{A2B} \]

where:

\[ |\Psi\rangle_{A1C} = \alpha |00\rangle + \beta |01\rangle + \gamma |10\rangle + \delta |11\rangle \]

represents the unknown quantum state shared between Alice and Charlie, and

\[ |\phi^+\rangle_{A2B} = \frac{1}{\sqrt{2}} \big(|00\rangle + |11\rangle \big) \]

represents the pre-shared Bell pair between Alice and Bob.

Expanding this tensor product, the full initial state of the system becomes:

\[ |\Psi\rangle_{A1C} \otimes |\phi^+\rangle_{A2B} = \frac{1}{\sqrt{2}} \big( \alpha |0000\rangle + \alpha |0011\rangle + \beta |0100\rangle + \beta |0111\rangle + \gamma |1000\rangle + \gamma |1011\rangle + \delta |1100\rangle + \delta |1111\rangle \big) \]

where the qubits are ordered as \( A1, A2, C, B \).

Objective

Alice will apply a series of quantum operations (CNOT and Hadamard), measure her qubits, and communicate the results to Bob, who will then apply correction unitaries to ensure that Bob and Charlie recover the original state \( |\Psi\rangle_{CB} \). The goal is to prove that the entangled state is successfully teleported, preserving its properties.

Stated below as:

Question 1

Alice, Bob, and Charlie share a quantum system where Alice holds two qubits (\(A1, A2\)), Bob holds one qubit (\(B\)), and Charlie holds one qubit (\(C\)). Alice wants to transfer an unknown quantum state \( |\Psi\rangle \) to Bob and Charlie using quantum teleportation.

Initial State of the System:

Alice’s first qubit \(A1\) and Charlie’s qubit \(C\) are entangled in an unknown quantum state:

\[ |\Psi\rangle_{A1C} = \alpha |00\rangle + \beta |01\rangle + \gamma |10\rangle + \delta |11\rangle \]

where \(\alpha, \beta, \gamma, \delta\) are complex coefficients satisfying the normalization condition:

\[ |\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1 \]

Additionally, Alice’s second qubit (\(A2\)) and Bob’s qubit (\(B\)) are entangled in a maximally entangled Bell state:

\[ |\phi^+\rangle_{A2B} = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle) \]

Thus, the total initial state of the system before Alice applies any operations is:

\[ |\Psi\rangle_{A1C} \otimes |\phi^+\rangle_{A2B} = \frac{1}{\sqrt{2}} \Big( \alpha |0000\rangle + \alpha |0011\rangle + \beta |0100\rangle + \beta |0111\rangle + \gamma |1000\rangle + \gamma |1011\rangle + \delta |1100\rangle + \delta |1111\rangle \Big) \]

Solution

Step 1: Alice Applies a CNOT Gate on Her Qubits

Alice applies a CNOT gate on her qubits \( A1 \) (control) and \( A2 \) (target). The CNOT operation flips the target qubit if the control qubit is \( |1\rangle \).

Applying CNOT to \( A1 \) (control) and \( A2 \) (target):

\[ \frac{1}{\sqrt{2}} \Big( \alpha |0000\rangle + \alpha |0011\rangle + \beta |0100\rangle + \beta |0111\rangle + \gamma |1100\rangle + \gamma |1111\rangle + \delta |1000\rangle + \delta |1011\rangle \Big) \]

Step 2: Alice Applies a Hadamard Gate on Her First Qubit (A1)

The Hadamard gate creates a superposition:

\[ H |0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), \quad H |1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) \]

Applying Hadamard on \( A1 \):

\[ \frac{1}{2} \Big[ |0\rangle \big( \alpha |000\rangle + \alpha |011\rangle + \beta |010\rangle + \beta |001\rangle + \gamma |110\rangle + \gamma |101\rangle + \delta |100\rangle + \delta |111\rangle \big) \]

\[ + |1\rangle \big( \alpha |000\rangle + \alpha |011\rangle + \beta |010\rangle + \beta |001\rangle - \gamma |110\rangle - \gamma |101\rangle - \delta |100\rangle - \delta |111\rangle \big) \Big] \]

This transformation entangles Alice’s qubits, preparing them for measurement.

Step 3: Alice Measures Her Qubits

Alice measures her two qubits (\(A1, A2\)) in the computational basis (\(|00\rangle, |01\rangle, |10\rangle, |11\rangle\)). The system collapses into one of four possible states, and Bob must apply a correction based on Alice’s result.

Possible outcomes and remaining states for Bob and Charlie:

1. If Alice measures \( |00\rangle \):
   [ \(|\Psi\rangle_{CB} = \alpha |00\rangle + \beta |01\rangle + \gamma |10\rangle + \delta |11\rangle\) ] → No correction needed.

2. If Alice measures \( |01\rangle \):
   [ \(|\Psi\rangle_{CB} = \alpha |00\rangle + \beta |01\rangle - \gamma |10\rangle - \delta |11\rangle\) ] → Bob applies \( Z \) correction.

3. If Alice measures \( |10\rangle \):
   [ \(|\Psi\rangle_{CB} = \alpha |10\rangle + \beta |11\rangle + \gamma |00\rangle + \delta |01\rangle\) ] → Bob applies \( X \) correction.

4. If Alice measures \( |11\rangle \):
   [ \(|\Psi\rangle_{CB} = \alpha |10\rangle + \beta |11\rangle - \gamma |00\rangle - \delta |01\rangle\) ] → Bob applies \( XZ \) correction.

Step 4: Bob’s Correction Based on Alice’s Measurement

Alice sends her measurement results (2 classical bits) to Bob:
- \( |00\rangle \) → No correction.
- \( |01\rangle \) → Apply \( Z \).
- \( |10\rangle \) → Apply \( X \).
- \( |11\rangle \) → Apply \( XZ \).

After Bob applies the necessary corrections, he and Charlie now share the original state \( |\Psi\rangle_{CB} \), successfully completing the quantum teleportation protocol.

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Conclusion

This process successfully transfers a quantum state using entanglement and classical communication:

1. Pre-shared entanglement between Alice & Bob.
2. Quantum operations (CNOT, Hadamard) applied by Alice.
3. Measurement of Alice’s qubits.
4. Classical communication of Alice’s measurement results to Bob.
5. Bob’s unitary correction operations based on the received classical bits.

Thus, the quantum state \( |\Psi\rangle \) is teleported from Alice to Bob and Charlie.